EXTENSIONS OF TAME ALGEBRAS AND FINITE GROUP SCHEMES OF DOMESTIC
REPRESENTATION TYPE
ROLF FARNSTEINER
Abstract. Let k be an algebraically closed field. Given an extension A : B of finite-dimensional kalgebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is
preserved under ascent and descent. These results are then used to show that principal blocks of finite
group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent
to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of
domestic representation type and the underlying group schemes are closely related to binary polyhedral
group schemes.
Introduction
Let k be an algebraically closed field. Given a finite-dimensional k-algebra A, we denote by modA the
category of finitely generated left A-modules. We will write [M ] for the isoclass of M ∈ modA . If modA
affords only finitely many isoclasses of indecomposable objects, then A is called representation-finite.
A fundamental result of Drozd [11] asserts that A is either tame or wild in the sense of the following
definition:
Definition. An algebra A is referred to as tame, if for every d > 0, there exist (A, k[T ])-bimodules
X1 , . . . , Xn(d) which are free k[T ]-modules of rank d, such that all but finitely many isoclasses of
indecomposable A-modules of dimension d are of the form [Xi ⊗k[T ] kλ ] for i ∈ {1, . . . , n(d)} and
λ : k[T ] −→ k.
We say that A is wild, if there exists an (A, khx, yi)-bimodule X, which is a finitely generated free
right khx, yi-module, such that the functor
mod khx, yi −→ mod A ; M 7→ X ⊗khx,yi M
preserves indecomposables and reflects isomorphisms.
The notion of wildness derives from the fact that the module category of a wild algebra A is at least
as complicated as that of any other algebra, rendering the classification of indecomposable A-modules a
rather hopeless endeavor.
Remark. In the definition of tameness, it suffices to require the parametrizing (A, k[T ])-bimodules
(Xi )1≤i≤n(d) of the d-dimensional indecomposable modules to be finitely generated over k[T ]. In that
case, each torsion submodule t(Xi ) of the k[T ]-module Xi is an (A, k[T ])-submodule, and there exist
non-zero polynomials fi ∈ k[T ] such that t(Xi )fi = (0). Let f := f1 f2 · · · fn(d) . Then the principal
open subset D(f ) := {λ ∈ Homalg (k[T ], k) ; λ(f ) 6= 0} is cofinite.
Let M be a d-dimensional indecomposable A-module. Assume that, with only finitely many exceptions,
we have [M ] = [Xi ⊗k[T ] kλ ] for some i ∈ {1, . . . , n(d)} and λ : k[T ] −→ k. If λ ∈ D(f ), then
Date: March 12, 2013.
2010 Mathematics Subject Classification. Primary 16G60, 14L15.
Supported by the D.F.G. priority program SPP1388 ‘Darstellungstheorie’.
1
2
ROLF FARNSTEINER
t(Xi )⊗k[T ]kλ = (0), and the right exactness of the tensor product implies that [M ] = [Yi⊗k[T ]kλ ], where
Yi := Xi /t(Xi ) is an (A, k[T ])-bimodule that is a free k[T ]-module of rank d.
This implies in particular that tameness is preserved under Morita equivalence. In [20] the authors
show that this even holds for stable equivalence.
Let A be a tame algebra. For each d > 0, we let µA (d) be the minimum of all possible numbers n(d)
occurring in the definition. We thus obtain a function µA : N −→ N0 , which we refer to as the growth
function of the tame algebra A. The bimodules X1 , . . . , Xn(d) define morphisms
fi,d : A1 −→ moddA ; λ 7→ Xi ⊗k[T ] kλ ,
with values in the variety moddA of d-dimensional A-modules. Accordingly, these bimodules provide continuous one-parameter families of indecomposable A-modules of dimension d. This fact notwithstanding,
there exist prominent examples of tame algebras, where a classification of the indecomposables has remained elusive. This has led to the definition of various subclasses of tame algebras, whose module
categories promise to be better behaved. In this paper, we are ultimately concerned with the following
class, introduced by Ringel [28]:
Definition. An algebra A is called domestic, if there exist (A, k[T ])-bimodules X1 , . . . , Xm that are free
of finite rank over k[T ] such that for every d > 0, all but finitely many isoclasses of d-dimensional
indecomposable A-modules are of the form [Xi ⊗k[T ] (k[T ]/(T −λ)j )] for some i ∈ {1, . . . , m}, j ∈ N
and λ ∈ k.
Remarks. (1) A group algebra kG of a finite group G is representation-infinite and domestic if and only
if char(k) = 2, and the Sylow 2-subgroups of G are Klein four groups, cf. [5, (4.4.4)]. The restricted
enveloping algebra U0 (sl(2)) of the restricted Lie algebra sl(2) over a field of characteristic p > 0 is
domestic.
(2) By the above remark, domesticity is preserved under Morita equivalence.
Finite groups and restricted Lie algebras are examples of finite group schemes. The purpose of this paper
is determine those finite k-group schemes G of characteristic p ≥ 3, whose algebra kG of measures has
domestic representation type. To that end, we study particular algebra extensions A : B, and show that
the more general notion of polynomial growth is preserved under ascent and descent.
Given a function f : N −→ N0 , we say that f has polynomial growth, if there exist c > 0 and n ≥ 0
such that f (d) ≤ cdn−1 for all d ≥ 1. The minimal number n ∈ N0 with this property is the rate of
growth γf of the function f . The following class of algebras was introduced by Skowroński [29]:
Definition. A tame algebra A is said to be of polynomial growth, if µA has polynomial growth. In that
case, we refer to γA := γµA as the rate of growth of the tame algebra A.
Remarks. Let A be tame.
(a) In view of Brauer-Thrall II (cf. [3, (IV.5)]), the algebra A is representation-finite if and only if A
is tame and γA = 0.
(b) Thanks to [8, (5.7)], the algebra A is domestic if and only if A is tame and γA ≤ 1.
(c) Suppose that A is of polynomial growth. If B is a k-algebra that is Morita equivalent to A, then
the above remark implies that B is also of polynomial growth.
FINITE GROUP SCHEMES OF DOMESTIC TYPE
3
(d) If A is of polynomial growth and I A is an ideal, then parametrizing (A, k[T ])-bimodules
X1 , . . . , XµA (d) induce parametrizing (A/I, k[T ])-bimodules (Xj /IXj )1≤j≤µA (d) , and the above
remark implies that A/I is of polynomial growth of rate γA/I ≤ γA .
(e) As will be shown in Section 4, group schemes of polynomial growth are of domestic representation
type.
Our main result provides a criterion guaranteeing that polynomial growth is preserved under passage
between the constituents of certain extensions A : B. We say that an extension A : B of k-algebras
is split, if B is a direct summand of the (B, B)-bimodule A. We refer to A : B as separable, if the
multiplication of A induces a split surjective morphism µ : A⊗B A −→ A of (A, A)-bimodules.
Theorem. Suppose that A : B is an extension of algebras.
(1) If A : B is split and A is of polynomial growth, then B is of polynomial growth, and γB ≤ γA+1.
(2) If A : B is separable and B is of polynomial growth, then A is of polynomial growth and
γA ≤ γB +1.
It is well-known that finite representation type is preserved when descending via a split extension or
ascending via a separable extension, cf. [27].
Our approach involves a geometric interpretation of growth functions that elaborates on a result by de
la Peña [25]. Section 1 provides the requisite tools which are then applied in Section 2, where the above
result and some refinements are established. The concluding Section presents examples and applications
concerning smash products and finite group schemes. In particular, we show that the principal block
B0 (G) of a finite group scheme of characteristic p ≥ 3 is representation-infinite and domestic if and
only if B0 (G) is Morita equivalent to a trivial extension of a radical square zero tame hereditary algebra.
Accordingly, one has a fairly good understanding of the module categories of these blocks (cf. [17,
(V.3.2)]).
1. Geometric Characterization of µA
Let k be an algebraically closed field. Throughout, A is assumed to be a finite-dimensional k-algebra.
Given n ∈ N, we denote by modnA the affine variety of n-dimensional A-modules. An element of modnA
is a homomorphism % : A −→ Endk (k n ) of k-algebras. The general linear group GLn (k) acts on modnA
via
(g.%)(a) := g ◦ %(a) ◦ g −1
∀ g ∈ GLn (k), % ∈ modnA , a ∈ A.
Note that the orbits under this action are just the isomorphism classes of the underlying A-modules.
Consequently, the subset indnA ⊆ modnA of n-dimensional indecomposable A-modules is GLn (k)-stable.
A subset C of a topological space X is locally closed, if it is the intersection of an open subset and a
closed subset of X. We say that C is constructible, if it is a finite union of locally closed sets. By virtue
of [7, (AG.1.3)], every constructible subset C of a noetherian topological space X contains a dense, open
subset of its closure. In particular, constructible subsets of varieties have this important property.
The following geometric criterion (cf. [25, (1.2)]), is based on Drozd’s Tame-Wild dichotomy [11]:
Proposition 1.1. The following statements are equivalent:
(1) The algebra A is tame.
(2) For every natural number d ∈ N there exists a closed subset Cd ⊆ moddA such that dim Cd ≤ 1
and inddA ⊆ GLd (k).Cd .
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ROLF FARNSTEINER
Suppose that A is tame with parametrizing bimodules X1 , . . . , XµA (d) for some d > 0. Let fi : A1 −→
moddA be the morphism determined by Xi , and put Vi := im fi . By definition, there exists a finite subset
Fd ⊆ moddA such that
µA (d)
[
inddA ⊆
GLd (k).Vi ∪ GLd (k).Fd .
i=1
We record the following elementary observation:
Lemma 1.2. Let V be a one-dimensional, irreducible variety. If C ⊆ V is constructible, then C is either
finite or cofinite.
Proof. Since C is constructible, it contains a dense open subset O of its closure C̄.
If dim C̄ = 0, then C is finite. Otherwise, dim C̄ = 1, so that O is dense and open in C̄ = V . Thus,
V rO is a proper, closed subset of V , whence dim V rO = 0. As a result, O and C are cofinite subsets
of V .
Lemma 1.3. Let d > 0. Suppose there exist one-dimensional closed, irreducible subsets V1 , . . . , V`(d) of
moddA , and a finite subset Fd ⊆ moddA such that
`(d)
inddA
⊆
[
GLd (k).Vi ∪ GLd (k).Fd ,
i=1
with `(d) being minimal subject to the above property. Then the following statements hold:
(1) |Vi ∩ GLd (k).x| < ∞
∀ x ∈ moddA , 1 ≤ i ≤ `(d).
d
(2) Vi ∩ indA is cofinite in Vi for every i ∈ {1, . . . , `(d)}.
S
(3) Ui := Vi ∩ [inddA r( `6=i GLd (k).V` )] is a dense, open subset of Vi for 1 ≤ i ≤ `(d).
Proof. (1) Let x be an element of moddA . By Chevalley’s theorem [16, (10.20)], Vi ∩ GLd (k).x is
a constructible subset of Vi , and therefore, by Lemma 1.2, finite or cofinite in Vi . If there exists an
element x0 ∈ moddA with Vi ∩ GLd (k).x0 cofinite in Vi , then there is a finite set Hd ⊆ moddA such that
Vi ⊆ GLd (k).x0 ∪ Hd , so that GLd (k).Vi ⊆ GLd (k).(Hd ∪ {x0 }). This contradicts the choice of `(d).
(2) Note that inddA is constructible (cf. [25, (1.1)]). If the constructible subset Vi ∩ inddA ⊆ Vi is
S
finite, then the GLd (k)-invariance of inddA yields inddA ⊆ `6=i GLd (k).V` ∪ GLd (k).(Fd ∪ (Vi ∩ inddA )),
which contradicts the minimality of `(d). Consequently, Vi ∩ inddA ⊆ Vi is a cofinite subset of Vi .
(3) Let i ∈ {1, . . . , `(d)}. By Chevalley’s Theorem [16, (10.20)], the image of GLd (k).V` of V` under
the multiplication GLd (k) × moddA −→ moddA is constructible. As inddA is constructible, it follows that
Ui is constructible. In view of Lemma 1.2, the set Ui is finite or cofinite in Vi .
By definition, we have
[
inddA ⊆ GLd (k).Ui ∪
GLd (k).V` ∪ GLd (k).Fd .
`6=i
By choice of `(d), the set Ui is not finite. Hence Ui is cofinite in Vi and thus a non-empty, open subset
of the one-dimensional irreducible variety Vi . This implies that Ui lies dense in Vi .
We require the following refinement of Proposition 1.1:
Proposition 1.4. The following statements are equivalent:
FINITE GROUP SCHEMES OF DOMESTIC TYPE
5
(1) There exists a function ` : N −→ N0 such that:
(a) For every d > 0 there exist one-dimensional irreducible closed subsets V1 , . . . , V`(d) ⊆ moddA ,
S`(d)
and a finite subset Fd ⊆ inddA such that inddA ⊆ i=1 GLd (k).Vi ∪ GLd (k).Fd , and
(b) `(d) is minimal subject to the properties listed in (a).
(2) The algebra A is tame with growth function µA = `.
Proof. (1) ⇒ (2) Thanks to Proposition 1.1, we know that A is tame. Let X1 , . . . , XµA (d) be (A, k[T ])bimodules giving rise to morphisms
fi : A1 −→ moddA
SµA (d)
such that inddA ⊆ GLd (k).( i=1
im fi ) ∪ GLd (k).Gd for some finite set Gd ⊆ inddA . Condition (b)
readily yields `(d) ≤ µA (d) for all d > 0.
Let d > 0. By assumption, there exists a finite subset Fd ⊆ inddA such that
`(d)
inddA ⊆
[
GLd (k).Vi ∪ GLd (k).Fd ,
i=1
with `(d) being minimal subject to this property. For each j ∈ {1, . . . , `(d)} we have
µA (d)
Vj ∩
inddA
⊆
[
(Vj ∩ GLd (k). im fi ) ∪ (Vj ∩ GLd (k).Gd ),
i=1
with each constituent of the union being constructible. By Lemma 1.3(1), the set Vj ∩ GLd (k).Gd is
finite, while Lemma 1.3(2) shows that Vj = Vj ∩ inddA . Hence there is an element ζ(j) ∈ {1, . . . , µA (d)}
such that
Vj = Vj ∩ GLd (k). im fζ(j) .
In particular, the constructible set Vj ∩ GLd (k). im fζ(j) is not finite. Thus, Lemma 1.2 provides a finite
set Rj with Vj = (Vj ∩ GLd (k). im fζ(j) ) ∪ Rj ⊆ GLd (k). im fζ(j) ∪ Rj , whence
GLd (k).Vj ⊆ GLd (k). im fζ(j) ∪ GLd (k).Rj .
As a result, there is a function ζ : {1, . . . , `(d)} −→ {1, . . . , µA (d)} such that
`(d)
inddA ⊆
[
`(d)
GLd (k). im fζ(j) ∪ GLd (k).(
j=1
[
Rj ∪ Fd ).
j=1
Hence all but finitely many isoclasses of d-dimensional indecomposable A-modules are of the form
[Xζ(j) ⊗k[T ] kλ ] for j ∈ {1, . . . , `(d)} and λ : k[T ] −→ k. Consequently, µA (d) ≤ | im ζ| ≤ `(d),
so that ` = µA is the growth function of the tame algebra A.
(2) ⇒ (1) Let d > 0 and consider the (A, k[T ])-bimodules X1 , . . . , XµA (d) that are provided by the
definition of tameness. If fi : A1 −→ moddA are the associated morphisms, then the family Vi :=
im fi (1 ≤ i ≤ µA (d)) together with some finite set Fd ⊆ moddA satisfies condition (a).
Suppose we also have
`(d)
[
d
indA ⊆
GLd (k).Wj ∪ GLd (k).Fd0 ,
j=1
Fd0
with
finite and Wj irreducible, closed and of dimension 1. The above arguments provide a function
ξ : {1, . . . , µA (d)} −→ {1, . . . , `(d)} such that
Vi = Vi ∩ GLd (k).Wξ(i) .
Consequently, ξ is injective, so that µA (d) ≤ `(d).
6
ROLF FARNSTEINER
For future reference, we record the following basic result:
Lemma 1.5. The map
m+n
n
sm,n : modm
; (M, N ) 7→ M ⊕ N
A × modA −→ modA
is a morphism of affine varieties.
2. Split Extensions
In the sequel, we are going to study the behavior of polynomial growth under split extensions. Throughout this section, A : B denotes an extension of finite dimensional k-algebras.
We begin with the following technical subsidiary result:
Lemma 2.1. Suppose that A : B is a split extension, with A being of polynomial growth. Then there
exists a function ` : N −→ N0 with the following properties:
(1) For every d > 0, there exist a finite subset Fd ⊆ inddB and (A, k[T ])-bimodules X1 , . . . , X`(d)
that are finitely generated free right k[T ]-modules such that for every V ∈ inddB rGLd (k).Fd
there exist j ∈ {1, . . . , `(d)} and λ : k[T ] −→ k with
(a) V being isomorphic to a direct summand of (Xj ⊗k[T ] kλ )|B , and
(b) Xj ⊗k[T ] kλ being isomorphic to a direct summand of A⊗B V .
(2) The function ` has polynomial growth of rate γ` ≤ γA +1.
(3) Further, if such ` is chosen so that `(d) is minimal subject to properties (a) and (b), then
|{Xj ⊗k[T ] kλ ; λ ∈ k} ∩ GLrj (k).x| < ∞
r
for all j ∈ {1, . . . , `(d)}, x ∈ modAj , rj := rkk[T ] (Xj ).
Proof. (1) If M is an (A, k[T ])-bimodule, we shall write M (λ) := M⊗k[T ]kλ for ease of notation. Since A
is tame, we can find, for every d > 0, a finite set Gd ⊆ inddA and (A, k[T ])-bimodules M1d , . . . MµdA (d) of
k[T ]-rank d such that every M ∈ inddA rGLd (k).Gd is isomorphic to Mid (λ) for some i ∈ {1, . . . , µA (d)}
and some λ : k[T ] −→ k.
S(dimk A)d
Gi as well as the finite-dimensional A-module Q :=
LConsider the finite set G(d) := i=d
M
.
By
the
Theorem
of
Krull-Remak-Schmidt,
the set
M ∈G(d)
Ud := {V ∈ inddB ; V is isomorphic to a direct summand of Q|B }
is the union of finitely many GLd (k)-orbits. Hence there exists a finite set Fd ⊆ inddB with
Ud = GLd (k).Fd .
Given V ∈
inddB rGLd (k).Fd ,
the splitting property of A : B provides a decomposition
(A⊗B V )|B ∼
=V ⊕W
of B-modules. There exist indecomposable A-modules N1 , . . . , Nq such that A ⊗B V ∼
= N1 ⊕ · · · ⊕
Nq , and the Theorem of Krull-Remak-Schmidt provides j ∈ {1, . . . , q} such that V is isomorphic to
a direct summand of Nj |B . Since dimk A ⊗B V ≤ (dimk A)d, we have Nj ∈ indtA for some t ∈
{d, . . . , (dimk A)d}. The assumption Nj ∈ GLt (k).Gt entails V ∈ Ud , a contradiction. Consequently,
d(dimk A)
Nj is isomorphic to some Mit (λ). As a result, the modules M1d , . . . , MµdA (d) , M1d+1 , . . . , MµA (d(dim
k A))
have the requisite properties.
FINITE GROUP SCHEMES OF DOMESTIC TYPE
7
(2) Given d > 0, we write n := dimk A and put {Mrs ; 1 ≤ s ≤ nd, 1 ≤ r ≤ µA (s)} =:
{X1 , . . . , X`(d) }. By assumption, there exists a constant c0 > 0 such that µA (d) ≤ c0 dγA −1 for all
d ≥ 1. Part (1) implies
`(d) ≤
nd
X
i=1
µA (i) ≤ c0
nd
X
iγA −1 ≤ c0 nd(nd)γA −1 = c0 nγA dγA .
i=1
Consequently, ` has polynomial growth of rate γ` ≤ γA +1.
r
(3) If the assertion is false, then there exist j ∈ {1, . . . , `(d)} and x0 ∈ modAj such that the constructible set Vj := {Xj ⊗k[T ] kλ ; λ ∈ k} ∩ GLrj (k).x0 is cofinite in {Xj ⊗k[T ] kλ ; λ ∈ k}. We write
r
{Xj ⊗k[T ] kλ ; λ ∈ k} = Vj ∪ Grj for some finite set Grj ⊆ modAj , so that
{Xj ⊗k[T ] kλ ; λ ∈ k} ⊆ GLrj (k).(Grj ∪ {x0 }).
For every M ∈ Grj ∪ {x0 }, we fix a decomposition of M |B into indecomposables, and consider the finite
L
subset Fd ⊆ inddB of d-dimensional indecomposable constituents of M ∈Gr ∪{x0 } M |B .
j
Let V be an element of inddB rGLd (k).(Fd ∪Fd ). Since V 6∈ GLd (k).Fd , there exist q ∈ {1, . . . , `(d)}
and λ : k[T ] −→ k such that
(a) V is a direct summand of some (Xq ⊗k[T ] kλ )|B , and
(b) Xq ⊗k[T ] kλ is a direct summand of A⊗B V .
In view of (a), the assumption q = j implies V ∈ GLd (k).Fd , a contradiction. Thus, q 6= j, which
contradicts the minimality of `(d).
Under additional assumptions, the growth rate of the function ` : N −→ N0 of Lemma 2.1 is bounded
by γA . For instance, if γA = 0, then A is representation-finite and γ` = 0. The following result provides
conditions that apply in the context of skew group algebras.
Let M be an A-module. Given an automorphism ζ ∈ Aut(A), we denote by M (ζ) the A-module with
underlying k-space M and action
a.m := ζ −1 (a).m
∀ a ∈ A, m ∈ M.
Below, we shall apply this twisting operation to bimodules, which we shall consider as modules over the
enveloping algebra A⊗k Aop .
Lr
Corollary 2.2. Let A : B be an extension of k-algebras such that A ∼
= i=1 Ai is an isomorphism of
(B, B)-modules with Ai ∼
= B (idB ⊗ζi ) for some ζi ∈ Aut(B). Suppose that A is of polynomial growth.
Then there exists a function ` : N −→ N0 with the following properties:
(1) For every d > 0, there exist a finite subset Fd ⊆ inddB and (A, k[T ])-bimodules X1 , . . . , X`(d)
that are finitely generated free right k[T ]-modules such that for every V ∈ inddB rGLd (k).Fd
there exist i ∈ {1, . . . , `(d)} and λ : k[T ] −→ k such that
(a) V is isomorphic to a direct summand of (Xi ⊗k[T ] kλ )|B , and
(b) Xi ⊗k[T ] kλ is a direct summand of A⊗B V .
(2) The function ` has polynomial growth of rate γ` ≤ γA .
Proof. Let V ∈ inddB . Our current assumption implies that
(A⊗B V )|B ∼
=
r
M
(Ai ⊗B V )|B ∼
=
i=1
r
M
V (ζi ) .
i=1
Lq
Returning to the proof of Lemma 2.1, we write A⊗B V = i=1 Ni as a sum of indecomposable modules
and obtain that ti := dimk Ni = aNi d for 1 ≤ i ≤ q, where 1 ≤ aNi ≤ r. The arguments of (2.1) now
8
ROLF FARNSTEINER
imply that the modules M1d , . . . , MµdA (d) , M12d , . . . , Mµ2dA (2d) , . . . , MµrdA (rd) have the requisite properties.
Consequently,
r
X
`(d) ≤
µA (id) ≤ rc0 (rd)γA −1 = rγA c0 dγA −1 ,
i=1
as desired.
Let V be a variety. Given x ∈ V , we denote by dimx V the local dimension of V at x. By definition,
dimx V is the maximum dimension of all irreducible components of V containing x.
Theorem 2.3. Suppose that A : B is a split extension. If A is of polynomial growth, then B is of
polynomial growth with γB ≤ γA +1.
Proof. Since the extension is split, every indecomposable B-module is a direct summand of an indecomposable A-module. According to [26, (4.2)], the algebra B is therefore tame. Thus, for each d > 0,
there are parametrizing families
fi : A1 −→ moddB
;
1 ≤ i ≤ µB (d).
We set Vi := im fi , and apply Proposition 1.4 and Lemma 1.3(1) to obtain
(†)
|imfi ∩ GLd (k).x| < ∞
∀ x ∈ moddB .
Now let X1 , . . . , X`(d) , Fd be the bimodules and the finite subset of inddB provided by Lemma 2.1,
respectively. For j ∈ {1, . . . , `(d)} let
r
gj : A1 −→ modAj ;
rj := rkk[T ] (Xj )
be the corresponding parametrizing morphisms. We put Wj := im gj and choose `(d) to be minimal
subject to properties (a) and (b) of Lemma 2.1.
Chevalley’s Theorem provides dense open subsets Oi ⊆ Vi and Uj ⊆ Wj with Oi ⊆ im fi and
Uj ⊆ im gj .
Given (i, j) ∈ {1, . . . , µB (d)}×{1, . . . , `(d)}, we let S(i, j) ⊆ im fi ×im gj be the set of all those pairs
r
(M, N ) ∈ moddB × modAj such that M is isomorphic to a direct summand of N |B and N is isomorphic
to a direct summand of A⊗B M . In view of Lemma 1.5 and the arguments of [25, p.183], this set is
constructible. We denote by π(i,j) : S(i, j) −→ Vi the restriction of the projection Vi × Wj −→ Vi .
(a) We have dim S(i, j) ≤ 1.
Since dim Vi = 1 = dim Wj , we have dim S(i, j) ≤ 2. Suppose equality to hold. Then S(i, j) = Vi × Wj
is an irreducible variety. Thus, Oi × Uj ⊆ im fi × im gj is an open, and thus dense, subset of S(i, j).
As S(i, j) is constructible, it also contains a dense, open subset of the irreducible variety S(i, j). Its
non-empty intersection with Oi × Uj will be denoted O.
−1
Let (M, N ) ∈ O and consider an element (V, W ) ∈ O ∩ π(i,j)
(π(i,j) (M, N )). Then M = V , M
is a direct summand of W |B , and W is direct summand of the A-module A⊗B M . The Theorem of
r
Krull-Remak-Schmidt provides a finite subset F ⊆ modAj such that
−1
O ∩ π(i,j)
(π(i,j) (M, N )) ⊆ {M } × (Uj ∩ GLrj (k).F).
−1
Thanks to Lemma 2.1(3), the set Uj ∩ GLrj (k).F is finite, so that O ∩ π(i,j)
(π(i,j) (M, N )) also has this
property.
FINITE GROUP SCHEMES OF DOMESTIC TYPE
9
−1
Let Z ⊆ π(i,j)
(π(i,j) (M, N )) be an irreducible component containing (M, N ). Then (M, N ) ∈ O ∩ Z,
so that O ∩ Z is a dense, open subset of Z. Since O ∩ Z is finite, we obtain dim Z = dim Z ∩ O = 0,
implying
−1
dim(M,N ) π(i,j)
(π(i,j) (M, N )) = 0
∀ (M, N ) ∈ O.
Hence the generic fiber of the morphism π(i,j) is zero-dimensional, so that dim Vi ≥ 2, a contradiction
cf. [23, (I.§8)].
S`(d)
(b) We have Vi = j=1 π(i,j) (S(i, j)).
Thanks to (†), the set Oi ∩GLd (k).Fd is finite. Thus, Oir(Oi ∩GLd (k).Fd ) is a dense open subset of the
one-dimensional irreducible variety Vi , and Lemma 1.3(2) implies that Oi0 := (Oi r(Oi ∩ GLd (k).Fd )) ∩
(Vi ∩ inddA ) enjoys the same property. Owing to Lemma 2.1(1), we have
`(d)
Oi0 ⊆
[
π(i,j) (S(i, j)),
j=1
whence
`(d)
Vi = Oi0 ⊆
[
π(i,j) (S(i, j)) ⊆ Vi ,
j=1
as desired.
Let f : X −→ Y be a continuous map between topological spaces, Q ⊆ X be a subset. Since
f (Q) ⊆ f (Q), we have
f (Q) = f (Q).
Accordingly, a morphism γ : S(i, j) −→ Vi is dominant if and only if γ(S(i, j)) = Vi .
(c) Let σ(i,j) : S(i, j) −→ Wj be induced by the projection Vi × Wj −→ Wj . If Z ⊆ S(i, j) is an
irreducible component such that π(i,j) : Z −→ Vi is a dominant morphism, so is σ(i,j) : Z −→ Wj .
Suppose the contrary, so that σ(i,j) : Z −→ Wj has a finite image. Thanks to (a) and the choice of
−1
Z, we have dim Z = 1. Setting r := min{dimz σ(i,j)
(σ(i,j) (z)) ; z ∈ Z}, we conclude from upper
semicontinuity of fiber dimension [23, (I.§8)] that
−1
O := {z ∈ Z ; dimz σ(i,j)
(σ(i,j) (z)) = r}
is a dense, open subset of the one-dimensional variety Z. On the other hand, the generic fiber dimension
−1
theorem implies dimz σ(i,j)
(σ(i,j) (z)) = dim Z−dim σ(i,j) (Z) = 1 on a non-empty, open subset of Z, so
that r = 1.
Since the morphism π(i,j) : Z −→ Vi is dominant, its image intersects the open subset Oi ⊆ im fi ⊆
−1
Vi . Thus, π(i,j)
(Oi ) is a dense, open subset of Z. The constructible set S(i, j) contains a dense, open
subset U of S(i, j) (cf. [7, (AG.1.3)]). The assumption U ∩ Z = ∅ implies that U is contained in the
union of the irreducible components of S(i, j) different from Z. Thus, U ( S(i, j), a contradiction. As
a result,
−1
Õ := π(i,j)
(Oi ) ∩ U ∩ O
is a dense, open subset of Z.
−1
Let x := (M, N ) be an element of Õ. If (P, Q) ∈ Õ ∩ σ(i,j)
(σ(i,j) (M, N )), then we have Q = N and
P is a direct summand of N |B . Consequently, the Theorem of Krull-Remak-Schmidt provides a finite
subset F ⊆ moddB such that
−1
Õ ∩ σ(i,j)
(σ(i,j) (M, N )) ⊆ (Oi ∩ GLd (k).F) × {N }.
10
ROLF FARNSTEINER
−1
Thanks to property (†) the right-hand set is finite. Since x ∈ Õ, the set Õ ∩ σ(i,j)
(σ(i,j) (x)) intersects
−1
every irreducible component X of σ(i,j)
(σ(i,j) (x)) containing x non-trivially. Let A be the set of these
components. Consequently,
−1
dimx σ(i,j)
(σ(i,j) (x)) = max dim X = max dim X ∩ Õ = 0,
X∈A
which contradicts the fact that x ∈ O.
X∈A
(d) Let ϕ : V −→ W be a dominant morphism of one-dimensional irreducible varieties. Then ϕ is
open.
Let ∅ 6= O ⊆ V be an open set. Then ϕ(O) is a constructible subset of a one-dimensional irreducible
variety and thus by Lemma 1.2 finite or cofinite. In the former case, we have
W = ϕ(V ) = ϕ(O) ⊆ ϕ(O) = ϕ(O),
a contradiction. Thus, ϕ(O) is cofinite and in particular open.
(e) Given j ∈ {1, . . . , `(d)}, the set
Γj := {i ∈ {1, . . . , µB (d)} ; Vi = π(i,j) (S(i, j))}
has cardinality |Γj | ≤ dimk A.
Let r ∈ Γj . By assumption, there exists an irreducible component Zr ⊆ S(r, j) such that the morphism
π(r,j) : Zr −→ Vr is dominant. Thanks to (c), we also have a dominant morphism σ(r,j) : Zr −→ Wj .
−1
6 π(r,j)
(inddB ∩ Vr ) ⊆ Zr
In view of Lemma 1.3(2), inddB ∩Vr is a dense, open subset of Vr , so that ∅ =
enjoys the same property. By constructibility of S(r, j), there exists a dense open subset Ũr ⊆ S(r, j)
of S(r, j). As Zr is a component of S(r, j), this set intersects Zr non-trivially. Put Ur := Vr ∩
S
inddB r( `6=r GLd (k).V` ). In view of (d) and Lemma 1.3(3),
−1
Or := σ(r,j) (Ũr ∩ π(r,j)
(Ur )) ⊆ σ(r,j) (S(r, j))
is a dense, open subset of Wj . Thus, Õ :=
T
r∈Γj
Or also has these properties.
−1
We fix an element s ∈ Γj . Since σ(s,j) : Zs −→ Wj is dominant, we have σ(s,j)
(Õ) 6= ∅. As Ũs meets
−1
Zs , it follows that Us0 := Ũs ∩ σ(s,j)
(Õ) ⊆ S(s, j) is a dense, open subset of the irreducible variety Zs .
0
In view of (a) and (d), Os := π(s,j) (Us0 ) is a dense open subset of Vs .
Thanks to Lemma 1.3(2), Vs ∩ inddB is a dense open subset of Vs . Its intersection with Os0 yields a
dense open subset Os00 ⊆ Os0 ∩ inddB of Vs . In particular, we have Vs = Os00 ∪ Fs for some finite subset
Fs ⊆ Vs .
Let M be an element of Os00 . Then M is indecomposable and M = π(s,j) (M, N ) with (M, N ) ∈ Us0 .
Thus, (M, N ) ∈ S(s, j) and N = σ(s,j) (M, N ) ∈ Or ⊆ σ(r,j) (S(r, j)) for every r ∈ Γj . Hence there
exists, for every r ∈ Γj , a B-module Mr ∈ Ur such that (Mr , N ) ∈ S(r, j).
The B-modules (Mr )r∈Γj are indecomposable direct summands of N |B . As each Mr belongs
L to Ur ,
the Mr are pairwise non-isomorphic. By the Theorem of Krull-Remak-Schmidt, the module r∈Γj Mr
is also a direct summand of N |B . Since dimk N ≤ dimk A⊗B Ms ≤ (dimk A)d, we obtain
d|Γj | ≤ dimk N ≤ d(dimk A),
so that |Γj | ≤ dimk A.
FINITE GROUP SCHEMES OF DOMESTIC TYPE
11
Let d > 0. Since each Vi is irreducible, claim (b) provides a map ω : {1, . . . , µB (d)} −→ {1, . . . , `(d)}
such that Vi = π(i,ω(i)) (S(i, ω(i)). We set Ω := im ω. There exists a finite set Fd0 ⊆ moddB such that
[ [
[ [
inddB ⊆
GLd (k).Vi ∪ GLd (k).Fd0 ⊆
GLd (k).Vi ∪ GLd (k).Fd0 .
t∈Ω i∈ω −1 (t)
t∈Ω i∈Γt
Thanks to (e) and Lemma 2.1, there exists c > 0 such that the number µB (d) of one-parameter families
is bounded by
|Ω|(max |Γj |) ≤ `(d)(dimk A) ≤ c(dimk A)dγA .
j∈Ω
Consequently, the algebra B has polynomial growth of rate γB ≤ γA +1.
The values of γA and γB may be arbitrarily far apart. Let A be a tame algebra. Then A : k is a split
extension with γk = 0.
The upper bound for γB depends on the knowledge of the indecomposable constituents of A⊗B V
containing V as a direct summand. The following result shows how a better understanding of induced
modules leads to improved estimates.
L
Corollary 2.4. Suppose that A : B is an extension of k-algebras such that A = ri=1 Ai is a direct sum
of (B, B)-modules with Ai ∼
= B (idB ⊗ζi ) for some ζi ∈ Aut(B). If A is of polynomial growth, then B is
of polynomial growth with γB ≤ γA .
Proof. Using Corollary 2.2 in the last paragraph of the above proof, we arrive at the stronger estimate. Examples. (1) The classical example of a split extension is the trivial extension A := B ⊕ M of B by a
(B, B)-bimodule M . In that case, B is also a factor algebra of A and our prefatory remarks show that
B inherits polynomial growth from A along with γB ≤ γA .
(2) Let G be a finite group, H ⊆ G be a subgroup, S ⊆ G be a set of right coset representatives of
H in G containing 1. Then
M
kG = kH ⊕ (
kHs)
s6=1
is a direct sum of (kH, kH)-bimodules, so that the extension kG : kH is split.
If H G is normal in G, then
M
kG =
kHs
s∈S
−1
is a decomposition of kH-bimodules, with each summand being isomorphic to kH (idkH ⊗s. ) , where
s.−1 denotes the conjugation by s−1 . Suppose that kG has polynomial growth. It now follows from
Corollary 2.4 that kH has polynomial growth with growth rate γkH ≤ γkG .
(3) Suppose that char(k) = p > 0. The following example shows that extensions defined by “group
algebras” of infinitesimal group schemes tend to behave differently. Let b ⊆ sl(2) be the Borel subalgebra
of upper triangular (2×2)-matrices of the restricted Lie algebra sl(2). The extension U0 (sl(2)) : U0 (b),
given by the associated restricted enveloping algebras, is not split: Let λ ∈ b∗ be the Steinberg weight,
so that U0 (sl(2))⊗U0 (b) kλ is the Steinberg module. This module is known to be projective, so that its
restriction (U0 (sl(2))⊗U0 (b) kλ )|U0 (b) is also projective. If U0 (sl(2)) : U0 (b) was split, then kλ would be
a direct summand of (U0 (sl(2))⊗U0 (b) kλ )|U0 (b) and hence also projective. Since b is not a torus, this is
impossible. Note that γU0 (sl(2)) = 1, while γU0 (b) = 0.
12
ROLF FARNSTEINER
3. Separable Extensions
In this section we show that polynomial growth is preserved under ascent via separable extensions.
Lemma 3.1. Let A : B be a separable extension, and suppose that B is tame of polynomial growth.
Then there exists a function ` : N −→ N0 with the following properties:
(1) For every d > 0 there exist a finite subset Fd ⊆ inddA and (B, k[T ])-bimodules Y1 , . . . , Y`(d)
that are free k[T ]-modules of rank ≤ d such that for every M ∈ inddA rGLd (k).Fd there exist
i ∈ {1, . . . , `(d)} and λ : k[T ] −→ k with
(a) M being isomorphic to a direct summand of (A⊗B Yi )⊗k[T ] kλ , and
(b) Yi ⊗k[T ] kλ being isomorphic to a direct summand of M |B .
(2) The function ` has polynomial growth of rate γ` ≤ γB +1.
(3) Further, if such ` is chosen so that `(d) is minimal subject to (a) and (b), then
|{Yi ⊗k[T ] kλ ; λ ∈ k} ∩ GLri (k).x| < ∞
for all i ∈ {1, . . . , `(d)}, x ∈ modrBi , ri := rkk[T ] (Yi ).
Proof. (1) Given j > 0, we let X1j , . . . , Xµj B (j) be the parametrizing (B, k[T ])-modules and Gj ⊆ indjB
be the finite set of exceptional modules.
Let d > 0, M ∈ inddA . Since the extension A : B is separable, we have M | A ⊗B M , and there
exists an indecomposable direct summand N of M |B such that M | A⊗B N . Hence N ∈ indjB for some
j ∈ {1, . . . , d}.
We consider the (B, k[T ])-bimodules {Xij ; 1 ≤ j ≤ d, 1 ≤ i ≤ µB (j)}. Let `(d) be the number
of these modules, which we denote Y1 , . . . , Y`(d) . By definition, each Yi is a free k[T ]-module of rank
rkk[T ] (Yi ) ≤ d.
Consider the finite set
d
[
Ud := {A⊗B N ; N ∈
Gj }.
j=1
For each X ∈ Ud we pick a decomposition into indecomposable constituents. Let Ud be the finite set of
these constituents and put Fd := inddA ∩Ud .
Let M ∈ inddA rGLd (k).Fd . By the above, there exist j ∈ {1, . . . , d} and an indecomposable direct
summand N | (M |B ) with dimk N = j, such that M is isomorphic to a direct summand of A⊗B N . The
assumption N ∈ GLj (k).Gj implies M ∈ GLd (k).Fd , a contradiction. It follows that N ∼
= Yi ⊗k[T ] kλ
for some i ∈ {1, . . . , `(d)}.
(2) Since B has polynomial growth, there exists c > 0 such that
`(d) =
d
X
j=1
µB (j) ≤ c
d
X
j γB −1 ≤ cddγB −1 ≤ cdγB .
j=1
As a result, we have γ` ≤ γB +1.
(3) Suppose there is x0 ∈ modrBi for some i ∈ {1, . . . , `(d)} such that the set Vi := {Yi ⊗k[T ] kλ ; λ ∈
k} ∩ GLri (k).x0 is infinite. Lemma 1.2 then shows that Vi is cofinite in {Yi ⊗k[T ] kλ ; λ ∈ k}. We write
{Yi ⊗k[T ] kλ ; λ ∈ k} = Vi ∪ Gri
for some finite set Gri ⊆ modrBi . For every N ∈ Gri ∪ {x0 }, we fix a decomposition of A⊗B N into
indecomposables and consider the finite subset Fd ⊆ inddA of d-dimensional indecomposable constituents.
Let M be an element of inddA rGLd (k).(Fd ∪ Fd ). Since M 6∈ GLd (k).Fd , M is a direct summand
of some (A⊗B Yj )⊗k[T ] kλ , while M 6∈ GLd (k).Fd ensures that j 6= i. This, however, contradicts the
minimality of `(d).
FINITE GROUP SCHEMES OF DOMESTIC TYPE
13
Given a finite group G that acts on a k-algebra B via automorphisms, we denote by B ∗ G the skew
group algebra, cf. [2, (III.4)]. For separable extensions of the form B∗G : B, the estimate for the growth
of the function ` may be strengthened.
In the following, we denote by bqc the least integer greater than or equal to q ∈ Q.
Lemma 3.2. Let G be a finite group acting on a k-algebra B via automorphisms. Suppose that B has
polynomial growth, and that p := char(k) - ord(G). Then there exists a function ` : N −→ N0 with the
following properties:
(1) For every d > 0 there exist a finite subset Fd ⊆ inddB∗G and (B, k[T ])-bimodules Y1 , . . . , Y`(d)
that are free k[T ]-modules of rank ≤ d such that for every M ∈ inddB∗G rGLd (k).Fd there exist
i ∈ {1, . . . , `(d)} and λ : k[T ] −→ k such that
(a) M is isomorphic to a direct summand of (B ∗G⊗B Yi )⊗k[T ] kλ , and
(b) Yi ⊗k[T ] kλ is isomorphic to a direct summand of M |B .
(2) The function ` has polynomial growth of rate γ` ≤ γB .
(3) Further, if such ` is chosen so that `(d) is minimal subject to (a) and (b), then
|{Yi ⊗k[T ] kλ ; λ ∈ k} ∩ GLri (k).x| < ∞
for all i ∈ {1, . . . , `(d)}, x ∈ modrBi , ri := rkk[T ] (Yi ).
Proof. We put A := B ∗G for ease of notation. Since p does not divide ord(G), the extension A : B is
separable (cf. Proposition 4.1.1 below).
Given j > 0, we let X1j , . . . , Xµj B (j) be parametrizing (B, k[T ])-bimodules, Gj ⊆ indjB be the finite
set of exceptional modules.
Let d > 0, M ∈ inddA . Then there exists an indecomposable direct summand N of M |B such that M
L
is a direct summand of A⊗B N . Since (A⊗B N )|B ∼
= g∈G N (g) , there exists a subset SM ⊆ G with
L
j
M |B ∼
= g∈SM N (g) . As a result dimk M = |SM | dimk N , and we conclude that N ∈ indB , where
d = rj and r ≤ ord(G) =: |G|. We consider the (A, k[T ])-bimodules
bd/rc
A⊗B Xi
;
1 ≤ r ≤ |G| , 1 ≤ i ≤ µB (bd/rc).
Let `(d) be the number of these modules. Since B has polynomial growth, there exists c > 0 such that
`(d) =
|G|
X
µB (bd/rc) ≤ c
r=1
|G|
X
bd/rcγB −1 ≤ c|G|dγB −1 .
r=1
As a result, we have γ` ≤ γB . The remaining assertions follow as in Lemma 3.1.
Theorem 3.3. Let A : B be a separable extension. If B is of polynomial growth, then A has polynomial
growth with γA ≤ γB +1.
Proof. Since the extension A : B is separable, every indecomposable A-module is a direct summand of
A⊗B N for some indecomposable B-module N . It thus follows from [26, (4.2)] that A is tame. The
arguments of Theorem 2.3 now apply mutatis mutandis. We indicate the requisite changes.
Let d > 0. There are parametrizing morphisms
fi : A1 −→ moddA
;
1 ≤ i ≤ µA (d).
We set Vi := im fi , and apply Proposition 1.4 and Lemma 1.3(1) to obtain
(†)
|imfi ∩ GLd (k).x| < ∞
∀ x ∈ moddB .
14
ROLF FARNSTEINER
Now let Y1 , . . . , Y`(d) , Fd be the (B, k[T ])-bimodules and the finite subset of inddB provided by Lemma
3.1, respectively. For j ∈ {1, . . . , `(d)} let
r
gj : A1 −→ modBj ;
rj := rkk[T ] (Xj ) ≤ d
be the corresponding parametrizing morphisms. We put Wj := im gj and choose `(d) to be minimal
subject to properties (a) and (b) of Lemma 3.1.
Chevalley’s Theorem provides dense open subsets Oi ⊆ Vi and Uj ⊆ Wj with Oi ⊆ im fi and
Uj ⊆ im gj .
Given (i, j) ∈ {1, . . . , µB (d)} × {1, . . . , `(d)}, we denote by S(i, j) ⊆ Vi × Wj the set of all pairs
(M, N ) ∈ im fi × im gj such that M is isomorphic to a direct summand of A ⊗B N and N is isomorphic
to a direct summand of M |B . In view of Lemma 1.5 and the arguments of [25, p.183], this set is
constructible. We denote by π(i,j) : S(i, j) −→ Vi the restriction of the projection Vi × Wj −→ Vi .
Assertions (a)-(d)L
of Theorem 2.3 can now be shown to hold in this context as well. In (e), we reach
the conclusion that r∈Γj Mr is a direct summand of A⊗B N . Since rj ≤ d this implies
d|Γj | ≤ dimk A⊗B N ≤ (dimk A)rj ≤ (dimk A)d.
The remaining arguments of Theorem 2.3 may be adopted verbatim.
4. Applications
4.1. Smash products. Our first application concerns the smash product of Hopf algebras. We refer
the reader to [22] for general facts concerning Hopf algebras. In the following, we P
shall write η and ε
for the antipode and co-unit of a Hopf algebra and use Sweedler notation ∆(h) = (h) h(1) ⊗h(2) for
calculations involving the comultiplication.
Let H be a finite-dimensional Hopf algebra, B be an H-module algebra. By definition, B is an
H-module with action (h, b) 7→ h.b such that
P
(a) h.(ab) = (h) (h(1) .a)(h(2) .b) for all h ∈ H, a, b ∈ B, and
(b) h.1 = ε(h)1 for all h ∈ H.
We consider the smash product B]H, with underlying k-space B ⊗k H and multiplication
X
(a⊗h).(b⊗h0 ) =
a(h(1) .b)⊗h(2) h0 ∀ h, h0 ∈ H, a, b ∈ B.
(h)
Recall that the canonical maps b 7→ b⊗1 and h 7→ 1⊗h are injective homomorphisms of algebras [22,
Chap.4]. In this section we study the extension B]H : B.
A Hopf algebra H is referred to as cosemisimple if its dual algebra H ∗ is semisimple. Recall that an
extension A : B of k-algebras is a free Frobenius extension of first kind, provided
(a) A is a finitely generated free B-module, and
(b) there exists an isomorphism ϕ : A −→ HomB (A, B) of (A, B)-bimodules.
In this case, the B-module A is known to afford dual bases {x1 , . . . , xn } and {y1 , . . . , yn } such that
ϕ(xi )(yj ) = δi,j . The map
n
X
cA:B : A −→ A ; b 7→
yi bxi ,
i=1
the so-called Gaschütz-Ikeda operator, does not depend on the choice of the bases.
Proposition 4.1.1. Let B be an H-module algebra. Then the following statements hold:
(1) B]H : B is a free Frobenius extension of first kind.
(2) If H is cosemisimple, then the extension B]H : B is split.
FINITE GROUP SCHEMES OF DOMESTIC TYPE
15
(3) If H is semisimple, then the extension B]H : B is separable.
Proof. (1) Let λ ∈ H ∗ be a non-zero left integral of H ∗ , that is, a linear form satisfying
(∗)
f ∗ λ = f (1) λ ∀ f ∈ H ∗ ,
P
where f ∗g(h) = (h) f (h(1) )g(h(2) ) denotes the convolution of f, g ∈ H ∗ . We define a k-linear map
π : B]H −→ B by means of
π(a ⊗ h) := a λ(h)
∀ a ∈ B, h ∈ H.
Note that π((a ⊗ 1).(b ⊗ h)) = π(ab ⊗ h) = abλ(h) = aπ(b ⊗ h) as well as
X
X
a(h(1) .b)λ(h(2) )
π(a(h(1) .b) ⊗ h(2) 1) =
π((a ⊗ h).(b ⊗ 1)) =
(h)
(h)
X
h(1) λ(h(2) )).b).
= a ((
(h)
Given f ∈ H ∗ we obtain, observing (∗),
f (λ(h)1) = f (1) λ(h) =
X
X
f (h(1) )λ(h(2) ) = f (
h(1) λ(h(2) )),
(h)
(h)
so that the last term above equals
a λ(h)b = π(a ⊗ h)b.
Since λ is a Frobenius homomorphism of the Frobenius algebra H (cf. [21]), there exist bases {x1 , . . . , xn };
{y1 , . . . , yn } of H such that λ(xi yj ) = δi,j for 1 ≤ i, j ≤ n. Observing
X
X
π((1 ⊗ x).(1 ⊗ y)) = π(
ε(x(1) )1B ⊗ x(2) y) =
ε(x(1) )λ(x(2) y)
(x)
(x)
X
= λ(
ε(x(1) )x(2) y) = λ(xy),
(x)
we obtain π((1 ⊗ xi ).(1 ⊗ yj )) = δi,j 1 ≤ i, j ≤ n.PBy definition, the set {1 ⊗ x1 , . . . , 1 ⊗ xn } is a
basis for the left B-module P
B]H. Suppose that 0 = ni=1 (1 ⊗ yi ).(bi ⊗ 1) for some bi ∈ B. Then we
have 0P= π((1 ⊗ xj ).0) = ni=1 π((1 ⊗ xj )(1 ⊗ yi ))bi = bj for 1 ≤ j ≤ n. As a result, the k-vector
space ni=1 (1 ⊗ yi ).B has dimension dimk B]H, so that {1 ⊗ y1 , . . . , 1 ⊗ yn } generates B]H as a right
B-module. The result now follows from [4, (1.2)].
(2) By assumption, the algebra H ∗ is semisimple. Recall that the left H-comodule H obtains the
structure of a right H ∗ -module via
X
h.ψ :=
ψ(h(1) )h(2)
∀ h ∈ H, ψ ∈ H ∗ .
(h)
H ∗ -submodule
Consequently, k1 is an
of H and thus possesses a direct complement V . By general
theory [22, (1.6.4)], V is a left subcomodule of H, so that ∆(V ) ⊆ H⊗kV . There results a vector space
decomposition
B]H = B ⊕ (B ⊗k V ),
which obviously is a decomposition of left B-modules. It remains to show that B ⊗k V is a right Bsubmodule of B]H. Given a, b ∈ B and v ∈ V , we have
X
(a ⊗ v).(b ⊗ 1) =
a(v(1) .b) ⊗ v(2) ∈ B ⊗k V,
(v)
as desired.
16
ROLF FARNSTEINER
(3) We denote by cH and cB]H:B the Gaschütz-Ikeda operators of the Frobenius extensions H : k and
B]H : B, respectively. The arguments of (1) imply
cB]H:B (1⊗h) =
n
X
(1⊗yi ).(1⊗ h).(1⊗xi ) = 1⊗ cH (h)
i=1
for every h ∈ H. Since H is semisimple, the Frobenius
Pextension H : k is separable, and there exists an
element z of the center of H such that 1 = cH (z) = ni=1 yi zxi , cf. [12, (2.6)]. Consequently,
1⊗1 = cB]H:B (1⊗z) = (1⊗z).cB]H:B (1⊗1) = cB]H:B (1⊗1).(1⊗z).
As cB]H:B (1⊗1) belongs to the centalizer CB]H:B (B), the element 1⊗z enjoys the same property, and
[12, (2.6)] (cf. also [18, (2.18)]) shows that the extension is separable.
Proposition 4.1.2. Let B be an H-module algebra. Then the following statements hold:
(1) If H is cosemisimple and B]H is of polynomial growth, then B is of polynomial growth with
γB ≤ γB]H +1.
(2) If H is semisimple and B is of polynomial growth, then B]H is of polynomial growth with
γB]H ≤ γB +1.
Proof. (1) Thanks to Proposition 4.1.1(2), the extension B]H : B is split. Consequently, Theorem 2.3
implies our result.
(2) In view of Proposition 4.1.1(3), the extension B]H : B is separable and the assertion follows from
Theorem 3.3.
Our next result refines [25, (3.3)].
Corollary 4.1.3. Let G be a finite group acting on a k-algebra B. Then the following statements hold:
(1) If B ∗G has polynomial growth, then B has polynomial growth with γB ≤ γB∗G .
(2) Suppose that p := char(k) - ord(G). Then B has polynomial growth if and only if B ∗G has
polynomial growth with γB∗G = γB .
L
−1
Proof. (1) Since B ∗G = g∈G Bg is a (B, B)-direct decomposition of B ∗G with Bg ∼
= B (idB ⊗g .) ,
the assertion is a direct consequence of Corollary 2.4.
(2) By replacing the reference to Lemma 3.1 by Lemma 3.2 in the proof of Theorem 3.3, we conclude
that B being of polynomial growth implies that B∗G has polynomial growth with γB∗G ≤ γB . Part (1)
provides the reverse direction as well as the reverse inequality.
A finite group scheme G is said to be linearly reductive, provided its algebra of measures kG is semisimple.
If B is an algebra, viewed as a functor R 7→ B ⊗k R from commutative k-algebras to k-algebras, then
there is the notion of G acting on B via automorphisms. This is quivalent to B being a kG-module
algebra. For the general theory of affine group schemes, we refer to [19, 31].
Corollary 4.1.4. Let G be a linearly reductive group scheme that operates on B via automorphisms. If
B has polynomial growth, then B]kG has polynomial growth of rate γB]kG ≤ γB .
FINITE GROUP SCHEMES OF DOMESTIC TYPE
17
Proof. By general theory [31, (6.8)], the group scheme G is a semi-direct product
G = G0 o Gred ,
with an infinitesimal, normal subgroup G0 and a reduced group scheme Gred . This implies that
kG = kG0 ∗G(k)
is a skew group algebra. Since G is linearly reductive, Nagata’s Theorem [9, (IV.§3.3.6)] ensures that G0
is diagonalizable and p - ord(G(k)).
We first assume that G is infinitesimal, so that G = D is diagonalizable. By van den Bergh’s Theorem
[6] (see also [22, p.167]), we have
(B]kD)]kD∗ ∼
= B ⊗k Endk (kD),
so that the two-fold smash product is of polynomial growth with rate γ(B]kD)]kD∗ = γB . Since D
is diagonalizable, kD∗ = kX(D) is the group algebra of the character group X(D) of D. Thus,
(B]kD)]kD∗ = (B]kD)∗X(D), and Corollary 4.1.3(1) shows that B]kD has polynomial growth of rate
γB]kD ≤ γB .
In the general case, we write G = DoG, where D is diagonalizable and G is finite with p - ord(G). By
the first part, the algebra C := B]kD is of polynomial growth with rate γC ≤ γB . Since B]kG ∼
= C∗G,
the assertion now follows from Corollary 4.1.3(2).
4.2. Blocks of group algebras. In preparation for our discussion of finite group schemes, we briefly
recall the relevant result for the “classical” case concerning finite groups. Recall that polynomial growth
and domesticity are preserved under Morita equivalence.
Let G be a finite group, B ⊆ kG be a block of the group algebra kG. In the sequel, we denote by
DB ⊆ G the defect group of B, a p-subgroup of G. If kG affords a tame block B, then p = 2 and DB
is dihedral, semidihedral, or generalized quaternion, cf. [5, (4.4.4)]. We illustrate the use of Theorem
2.3 via the following well-known result, which underscores the scarcity of representation-infinite blocks
of polynomial growth:
Theorem 4.2.1. Suppose that char(k) = 2. Let B ⊆ kG be a representation-infinite block. Then the
following statements are equivalent:
(1) The block B is of polynomial gowth.
(2) DB ∼
= Z/(2)×Z/(2).
(3) B is Morita equivalent to k(Z/(2)×Z/(2)), kA4 , or to the principal block of kA5 .
(4) The block B is domestic.
Proof. (1) ⇒ (2) We put V4 := Z/(2)×Z/(2). Let CG (DB ) be the centralizer of DB in G and consider
the subgroup H := DB CG (DB ). In view of Brauer’s First Main Theorem (cf. [1, Thm.2,p.103] and its
succeeding remark), there exists a block b ⊆ B of kH such that the extension B : b is split. Theorem 2.3
now shows that b is of polynomial growth. Thanks to [5, (6.4.6)], the algebras b and kDB are Morita
equivalent, so that kDB is of polynomial growth. Since B is representation-infinite, the group DB is not
cyclic, see [5, (6.5)]. Hence [29, Prop.1] implies DB ∼
= V4 .
(2) ⇒ (3) Let e be the inertial index of B. General theory implies e ∈ {1, 3}. If e = 1, then [5,
(6.6.1)] provides an isomorphism B ∼
= Matn (kV4 ). If e = 3, then [5, (6.6.3)] shows that B is Morita
equivalent to the group algebra kA4 of the alternating group A4 on 4 letters, or to the principal block
of kA5 .
(3) ⇒ (4) Since V4 is the Sylow-2-subgroup of V4 , A4 and A5 , [5, (4.4.4)] implies that B is domestic.
(4) ⇒ (1) Trivial.
18
ROLF FARNSTEINER
4.3. Group schemes of domestic representation type. We shall use our results to determine (up to
a linearly reductive normal subgroup) all finite k-group schemes G over an algebraically closed field k of
odd characteristic, whose algebra kG of measures is representation-infinite and domestic. As in the case
of finite groups, non-domestic group schemes do not have polynomial growth.
Let (Λ, ε) be an augmented k-algebra. There is a unique block B0 (Λ) such that ε(B0 (Λ)) 6= (0).
This block is referred to as the principal block of Λ. If the finite group G acts on Λ via automorphisms
of augmented algebras, then Λ∗G is augmented with augmentation εG satisfying
εG (λg) = ε(λ)
∀ λ ∈ Λ, g ∈ G.
Lemma 4.3.1. There exists a (Λ, Λ)-bimodule X such that B0 (Λ∗G) = B0 (Λ) ⊕ X.
Proof. This follows directly from the proof of [13, (5.1.2)].
Let Λ be a k-algebra. The split extension T (Λ) := Λ n Λ∗ of Λ by its dual bimodule Λ∗ is referred
to as the trivial extension of Λ. We shall apply this construction to radical square zero tame hereditary
algebras. By definition, such an algebra is of the form kQ, where the quiver Q is a Euclidean diagram
of type Ãn , D̃n or Ẽ6,7,8 with an orientation, such that there are no paths of length 2. (For Ãn this is
possible if and only if n is odd.)
We denote by Z the center of the algebraic group scheme SL(2). Given a finite subgroup scheme
G ⊆ SL(2), we put P(G) := G/(G ∩ Z). The unique largest linearly reductive normal subgroup scheme
of a given finite group scheme G will be denoted Glr .
Our final result characterizes the representation-infinite finite algebraic groups of domestic representation type, showing their close connection with binary polyhedral groups and tame hereditary algebras.
Recall that a finite linearly reductive subgroup scheme of SL(2) is referred to as a binary polyhedral
group scheme. We denote by SL(2)1 the first Frobenius kernel of SL(2).
Theorem 4.3.2. Let G be a finite group scheme over an algebraically closed field k of characteristic
p ≥ 3. Then the following statements are equivalent:
(1) The principal block B0 (G) is representation-infinite and of polynomial growth.
(2) There exists a radical square zero tame hereditary algebra Λ such that B0 (G) is Morita equivalent
to T (Λ).
(3) The Hopf algebra kG is representation-infinite and domestic.
(4) The principal block B0 (G) is representation-infinite and domestic.
(5) There exists a binary polyhedral group scheme G̃ ⊆ SL(2) such that G/Glr ∼
= P(SL(2)1 G̃).
Proof. Let G := G(k) be the finite group of k-rational points. By general theory, we have
kG = kG0 ]kG = kG0 ∗G.
(1) ⇒ (2) In view of the above, Lemma 4.3.1 implies that the extension B0 (G) : B0 (G0 ) is split. Theorem
2.3 thus shows that the algebra B0 (G0 ) is of polynomial growth. Thanks to [14, (6.1)], B0 (G0 ) is
domestic. By [15, (3.1)], B0 (G0 ) is not representation-finite and [14, (5.1)] implies that the center
Cent(G0 ) is diagonalizable. The assertion now follows from [13, (7.3.1)].
(2) ⇒ (3) The algebra Λ is known to be representation-infinite and domestic [10, 24]. Thanks to
[30], the trivial extension T (Λ) also enjoys this property. It follows that the algebra B0 (G) is domestic
and of infinite representation type. As before, we see that B0 (G0 ) also has these properties, and [14,
(6.1)] implies the domesticity of kG0 . Since B0 (G) is not representation-finite, [13, (6.2.1)] ensures that
p does not divide ord(G). It now follows from Corollary 4.1.3(2) that kG = kG0 ∗G is of polynomial
growth with rate γkG = γkG0 ≤ 1. Hence kG is domestic and of infinite representation type.
(3) ⇒ (4) This follows from [15, (3.1)].
FINITE GROUP SCHEMES OF DOMESTIC TYPE
19
(4) ⇒ (5) Let T ⊆ SL(2) be the standard torus of diagonal (2 × 2)-matrices of determinant 1 and put
:= G/Glr . Then we have G0lr = ek , while [13, (1.1)] implies that B0 (G0 ) ∼
= B0 (G) is domestic. Since
0
Glr = ek , a consecutive application of [13, (1.2)] and [14, (6.1)] provides an isomorphism G00 ∼
= SL(2)1 Tr
for some r ≥ 1. According to [13, (6.2.1)], the prime p does not divide ord(G0 (k)), so that [13, (1.2)]
yields CG0 := CentG0red (G00 ) = ek . Consequently, CentG0 (G00 ) = ek , and our assertion follows from [13,
(7.1.2)].
(5) ⇒ (1) Let G̃ ⊆ SL(2) be a binary polyhedral group scheme. We consider the semidirect product
G0
G := SL(2)1 o G̃.
Since G̃ is linearly reductive, Corollary 4.1.4 guarantees that kG ∼
= k SL(2)1 ]k G̃ is domestic. It follows
that the factor algebra k SL(2)1 G̃ of kG is domestic. Consequently, B0 (P(SL(2)1 G̃)) ∼
= B0 (SL(2)1 G̃)
enjoys the same property. Let cxG (k) be the complexity of the trivial G-module, that is, the polynomial
rate of growth of a minimal projective resolution of k. If B0 (SL(2)1 G̃) is representation-finite, then 1 ≥
cxSL(2)1 G̃ (k) ≥ cxSL(2)1 (k) = 2, a contradiction. As a result, the principal block B0 (G) ∼
= B0 (G/Glr ) ∼
=
B0 (P(SL(2)1 G̃)) (cf. [13, (1.1)]) is of polynomial growth.
Remark. The binary polyhedral group schemes are completely understood, see [13, (3.3)]. They are
determined by their McKay quivers, which in turn give rise to the Ext-quivers of the principal blocks
B0 (P(SL(2)1 G̃).
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Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098
Kiel, Germany.
E-mail address: [email protected]